Question about normal distribution curves

Question

I’m in AP Statistics, and a question came up that is about two normal distributions. One is the height of men, with a mean of 69.5 and a standard deviation of 4, and the other is the height of women, with a mean of 65 and a standard deviation of 3. Is there a way I can overlap the two distributions and look at their areas to calculate the probability a man is taller than a woman? I’m not sure how to do it and I’m wondering if it is possible.

Answer 1

Let $X$ be a random variable representing a man's height and let $Y$ be a random variable representing a woman's height.

Now let $Z = X - Y$ or the difference in height between the man and the woman. You are effectively asking $P(X>Y)$ which is the same as asking $P(X-Y>0)$ or $P(Z>0)$.

The "normal difference distribution" (i.e. the distribution of $X-Y$) where $X \sim N(69.5, 4^2)$ and $Y \sim N(65, 3^2)$ is normally distributed as $N(69.5-65, 4^2+3^2)$ or $N(4.5, 25)$. You can integrate over the Normal PDF from $0$ to $\infty$ now in order to find $P(X-Y>0)$.

Overlaying the PDFs and integrating will tell you things like "there is a 60% chance that men are between $x$ and $y$ feet tall while there is only a 30% chance a woman will be between $x$ and $y$ feet tall", but it won't tell you anything about the probability a man will actually be taller than a woman. To figure that out, you need the joint distribution of $X$ and $Y$ or you need to find the distribution of $X-Y$, which I did above.